Let us briefly comment on some relationships with the twistor space constructions
in the literature, following mainly [1]}, [2]}, [3]}, [4]}, [5]}.
(We are only concerned with projective twistor space.)
The original construction of twistor space was done by Penrose in [1]}, where it is called non-linear graviton
and is specific to complex (anti-)self-dual 4-manifolds.
The non-linear graviton is a 3-complex-dimensional manifold
and its points are totally null surfaces in the (A)SD 4-manifold.
The precise relationship between the two spaces can be encoded in a double fibration, where
the correspondence space is the projective spin bundle.
For real 4-manifolds, one can, according to signature, introduce different `real structures' (involutions)
on the twistor space of the complex 4-manifold;
but there are also other routes to construct the twistor space of a real 4-manifold.
